graph the exponential function.\n$f(x)=3^{-x}$\nplot five points on the graph of the function, and also draw…

Explanation:

Step 1: Rewrite the function

Rewrite ( f(x) = 3^{-x} ) as ( f(x) = \left(\frac{1}{3}\right)^x ) for clarity.

Step 2: Identify the horizontal asymptote

As ( x \to \infty ), ( \left(\frac{1}{3}\right)^x \to 0 ), so the horizontal asymptote is ( y = 0 ).

Step 3: Calculate five points

• When ( x = -2 ): ( f(-2) = \left(\frac{1}{3}\right)^{-2} = 9 ) → Point: ((-2, 9))
• When ( x = -1 ): ( f(-1) = \left(\frac{1}{3}\right)^{-1} = 3 ) → Point: ((-1, 3))
• When ( x = 0 ): ( f(0) = \left(\frac{1}{3}\right)^0 = 1 ) → Point: ((0, 1))
• When ( x = 1 ): ( f(1) = \left(\frac{1}{3}\right)^1 = \frac{1}{3} ) → Point: ((1, \frac{1}{3}))
• When ( x = 2 ): ( f(2) = \left(\frac{1}{3}\right)^2 = \frac{1}{9} ) → Point: ((2, \frac{1}{9}))

Step 4: Plot points and asymptote

Plot the points ((-2, 9)), ((-1, 3)), ((0, 1)), ((1, \frac{1}{3})), ((2, \frac{1}{9})) and draw the horizontal asymptote ( y = 0 ).

Answer:

Five points: ((-2, 9)), ((-1, 3)), ((0, 1)), ((1, \frac{1}{3})), ((2, \frac{1}{9})); Asymptote: ( y = 0 )